A032005 -id:A032005 - cp's OEIS frontend

A032006 "AFK" (ordered, size, unlabeled) transform of 2,1,1,1,...

1, 2, 1, 5, 5, 7, 19, 21, 33, 41, 101, 109, 175, 231, 321, 623, 761, 1087, 1495, 2109, 2661, 4985, 5849, 8557, 11251, 15831, 20373, 27743, 44357, 55135, 76123, 101373, 136689, 178673, 240125, 303997, 475183, 578271, 793809, 1024991, 1387985, 1763719, 2363671

Offset: 0

Christian G. Bower

nonn

Number of compositions of n into distinct parts if there are 2 kinds of part 1. a(3) = 5: 3, 21, 21', 12, 1'2.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
C. G. Bower, Transforms (2)

Cf. A032303.

Cf. A032005, A032007, A032008.

b:= proc(n, i, p) option remember;
      `if`(n=0, p!, `if`(i<1, 0, b(n, i-1, p)+
      `if`(i>n, 0, `if`(n=1, 2, 1)*b(n-i, i-1, p+1))))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..45);  # Alois P. Heinz, Sep 05 2015

b[n_, i_, p_] := b[n, i, p] = If[n == 0, p!, If[i < 1, 0, b[n, i - 1, p] + If[i > n, 0, If[n == 1, 2, 1]*b[n - i, i - 1, p + 1]]]];
a[n_] := b[n, n, 0];
a /@ Range[0, 40] (* Jean-François Alcover, Sep 11 2019, after Alois P. Heinz *)

seq(n)={apply(p->subst(serlaplace(p), y, 1), Vec((1 + 2*x*y)*prod(k=2, n, 1 + x^k*y + O(x*x^n))))} \\ Andrew Howroyd, Jun 21 2018

a(0)=1 prepended by Alois P. Heinz, Sep 05 2015

A032007 "AFK" (ordered, size, unlabeled) transform of 1,2,3,4,...

Original entry on oeis.org

1, 1, 2, 7, 10, 25, 68, 111, 208, 435, 1218, 1773, 3586, 6077, 12156, 31961, 47624, 86825, 151962, 265525, 469610, 1242607, 1750108, 3217663, 5263928, 9205197, 14713474, 26440503, 63610938, 90877893, 159360628, 258871127, 431309688, 687140639, 1134231986

Offset: 0

Christian G. Bower

nonn

Sum of products of parts in all compositions of n into distinct parts. - Vladeta Jovovic, Feb 21 2005

Number of compositions of n into distinct parts if there are i kinds of part i. a(3) = 7: 3, 3', 3'', 21, 2'1, 12, 12'.

Alois P. Heinz, Table of n, a(n) for n = 0..5000
C. G. Bower, Transforms (2)

Cf. A022629.

Cf. A032005, A032006, A032008.

b:= proc(n, i, p) option remember;
      `if`(n=0, p!, `if`(i<1, 0, b(n, i-1, p)+
      `if`(i>n, 0, i*b(n-i, i-1, p+1))))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..40);  # Alois P. Heinz, Sep 05 2015

b[n_, i_, p_] := b[n, i, p] = If[n == 0, p!, If[i < 1, 0, b[n, i - 1, p] + If[i > n, 0, i*b[n - i, i - 1, p + 1]]]];
a[n_] := b[n, n, 0];
a /@ Range[0, 40] (* Jean-François Alcover, Sep 11 2019, after Alois P. Heinz *)

seq(n)={apply(p->subst(serlaplace(p), y, 1), Vec(prod(k=1, n, 1 + k*x^k*y + O(x*x^n))))} \\ Andrew Howroyd, Jun 21 2018

a(0)=1 prepended by Alois P. Heinz, Sep 05 2015

A032008 "AFK" (ordered, size, unlabeled) transform of 1,3,5,7,...

Original entry on oeis.org

1, 1, 3, 11, 17, 53, 161, 285, 569, 1459, 4699, 7177, 15631, 28229, 66883, 211311, 319929, 627705, 1163049, 2150209, 4422225, 14320583, 20392019, 39962165, 68618087, 126643545, 212615483, 433704811, 1312156393, 1864959757, 3502343041, 5919183485, 10364053441

Offset: 0

Christian G. Bower

nonn

Number of compositions of n into distinct parts if there are (2i-1) kinds of part i. a(3) = 11: 3, 3', 3'', 3''', 3'''', 21, 2'1, 2''1, 12, 12', 12''. - Alois P. Heinz, Sep 05 2015

Alois P. Heinz, Table of n, a(n) for n = 0..4000
C. G. Bower, Transforms (2)

Cf. A032005, A032006, A032007.

b:= proc(n, i, p) option remember;
      `if`(n=0, p!, `if`(i<1, 0, b(n, i-1, p)+
      `if`(i>n, 0, (2*i-1)*b(n-i, i-1, p+1))))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..40);  # Alois P. Heinz, Sep 05 2015

b[n_, i_, p_] := b[n, i, p] = If[n == 0, p!, If[i < 1, 0, b[n, i - 1, p] + If[i > n, 0, (2 i - 1)*b[n - i, i - 1, p + 1]]]];
a[n_] := b[n, n, 0];
a /@ Range[0, 40] (* Jean-François Alcover, Sep 11 2019, after Alois P. Heinz *)

seq(n)={apply(p->subst(serlaplace(p), y, 1), Vec(prod(k=1, n, 1 + (2*k-1)*x^k*y + O(x*x^n))))} \\ Andrew Howroyd, Jun 21 2018

a(0)=1 prepended by Alois P. Heinz, Sep 05 2015

A032043 "BFK" (reversible, size, unlabeled) transform of 2,2,2,2...

Original entry on oeis.org

1, 2, 2, 6, 6, 10, 34, 38, 62, 90, 306, 334, 574, 794, 1226, 3390, 4014, 6370, 9130, 13598, 18470, 48114, 55290, 88966, 123238, 184178, 245714, 358710, 771990, 937210, 1427698, 1992662, 2882894, 3895626, 5533314, 7318270, 14935246

Offset: 0

Christian G. Bower

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..1000
C. G. Bower, Transforms (2)

a(n) = (A032005(n) + 2)/2.

seq(n)={apply(p->subst(serlaplace(p + polcoeff(p,1)), y, 1)/2, Vec(y-1+prod(k=1, n, 1 + 2*x^k*y + O(x*x^n))))} \\ Andrew Howroyd, Jun 21 2018

a(0)=1 prepended by Andrew Howroyd, Jun 21 2018

A355387 Number of ways to choose a distinct subsequence of an integer composition of n.

Original entry on oeis.org

1, 2, 5, 14, 37, 98, 259, 682, 1791, 4697, 12303, 32196, 84199, 220087, 575067, 1502176, 3923117, 10244069, 26746171, 69825070, 182276806, 475804961, 1241965456, 3241732629, 8461261457, 22084402087, 57640875725, 150442742575, 392652788250, 1024810764496

Offset: 0

Gus Wiseman, Jul 04 2022

nonn

By "distinct" we mean equal subsequences are counted only once. For example, the pair (1,1)(1) is counted only once even though (1) is a subsequence of (1,1) in two ways. The version with multiplicity is A025192.

			The a(3) = 14 pairings of a composition with a chosen subsequence:
  (3)()     (3)(3)
  (21)()    (21)(1)   (21)(2)    (21)(21)
  (12)()    (12)(1)   (12)(2)    (12)(12)
  (111)()   (111)(1)  (111)(11)  (111)(111)

Christian Sievers, Table of n, a(n) for n = 0..2000

For partitions we have A000712, composable A339006.
The homogeneous version is A011782, without containment A000302.
With multiplicity we have A025192, for partitions A070933.
The strict case is A032005.
The case of strict subsequences is A236002.
The composable case is A355384, homogeneous without containment A355388.
A075900 counts compositions of each part of a partition.
A304961 counts compositions of each part of a strict partition.
A307068 counts strict compositions of each part of a composition.
A336127 counts compositions of each part of a strict composition.
Cf. A011782, A022811, A032020, A063834, A133494, A181591, A323583, A331330, A336128, A336130, A336139, A355382, A355383.

Table[Sum[Length[Union[Subsets[y]]],{y,Join@@Permutations/@IntegerPartitions[n]}],{n,0,6}]

lista(n)=my(f=sum(k=1,n,(x^k+x*O(x^n))/(1-x/(1-x)+x^k)));Vec((1-x)/((1-2*x)*(1-f))) \\ Christian Sievers, May 06 2025

G.f.: (1-x)/((1-2*x)*(1-f)) where f = Sum_{k>=1} x^k/(1-x/(1-x)+x^k) is the generating function for A331330. - Christian Sievers, May 06 2025

a(16) and beyond from Christian Sievers, May 06 2025

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A032006 "AFK" (ordered, size, unlabeled) transform of 2,1,1,1,...

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A032007 "AFK" (ordered, size, unlabeled) transform of 1,2,3,4,...

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A032008 "AFK" (ordered, size, unlabeled) transform of 1,3,5,7,...

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A032043 "BFK" (reversible, size, unlabeled) transform of 2,2,2,2...

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A355387 Number of ways to choose a distinct subsequence of an integer composition of n.

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